Erdős Problem #142 — open problem in number theory

Erdős Problem #142

Prove an asymptotic formula for r_k(N), the largest possible size of a subset of 1, …, N that does not contain any non-trivial k-term arithmetic progression. That is, find f_k with r_k(N) / f_k(N) → 1 as N → ∞.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-erdos-142,
  title        = {Erdős Problem #142},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-142}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Prove an asymptotic formula for $r_{k} (N)$ , the largest possible size of a subset of ${1, \dots, N}$ that does not contain any non-trivial $k$ -term arithmetic progression. That is, find $f_{k}$ with $r_{k} (N) / f_{k} (N) \to 1$ as $N \to \infty$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«142». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_142 (k : ℕ) : (fun N => (r k N : ℝ)) ~[atTop] (answer(sorry) : ℕ → ℝ)

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already — check erdosproblems.com/142. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

erdos_142.variants.lower — Show that r_k(N) = o_k(N / log N), where r_k(N) the largest possible size of a subset of 1, …, N that does not contain any non-trivial k-term arithmetic…
erdos_142.variants.isTheta — Determine the order of magnitude of r_k(N), the largest possible size of a subset of 1, …, N that does not contain any non-trivial k-term arithmetic…
erdos_142.variants.three — Prove an asymptotic formula for r_3(N), the largest possible size of a subset of 1, …, N that does not contain any non-trivial 3-term arithmetic progression.

References

erdosproblems.com/142

Source and licence

Imported from Formal Conjectures (Erdős problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.